Réorganisation de la librairie des réels

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@3311 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 4 insertions, 66 deletions

diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v
index 7386cbb12..031870b93 100644
--- a/theories/Reals/NewtonInt.v
+++ b/theories/Reals/NewtonInt.v
@@ -8,22 +8,15 @@
  
 (*i $Id$ i*)
 
-Require Rbase.
-Require Rbasic_fun.
-Require DiscrR.
-Require Rderiv.
+Require RealsB.
+Require Rfunctions.
+Require SeqSeries.
 Require Rtrigo.
-Require Ranalysis1.
-Require R_sqrt.
-Require Ranalysis4.
-Require Rtopology.
-Require TAF.
+Require Ranalysis.
 
 (*******************************************)
 (*            Newton's Integral            *)
 (*******************************************)
- 
-Definition antiderivative [f,g:R->R;a,b:R] : Prop := ((x:R)``a<=x<=b``->(EXT pr : (derivable_pt g x) | (f x)==(derive_pt g x pr)))/\``a<=b``.
 
 Definition Newton_integrable [f:R->R;a,b:R] : Type := (sigTT ? [g:R->R](antiderivative f g a b)\/(antiderivative f g b a)).
 
@@ -62,61 +55,6 @@ Lemma NewtonInt_P3 : (f:R->R;a,b:R;X:(Newton_integrable f a b)) (Newton_integrab
 Unfold Newton_integrable; Intros; Elim X; Intros g H; Apply existTT with g; Tauto.
 Defined.
 
-Definition constant_D_eq [f:R->R;D:R->Prop;c:R] : Prop := (x:R) (D x) -> (f x)==c.
-
-(* If f has a null derivative in ]a,b[ and is continue in [a,b], *)
-(* then f is constant on [a,b] *)
-Lemma null_derivative_loc : (f:R->R;a,b:R;pr:(x:R)``a<x<b``->(derivable_pt f x)) ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ((x:R;P:``a<x<b``)(derive_pt f x (pr x P))==R0) -> (constant_D_eq f [x:R]``a<=x<=b`` (f a)).
-Intros; Unfold constant_D_eq; Intros; Case (total_order_T a b); Intro.
-Elim s; Intro.
-Assert H2 : (y:R)``a<y<x``->(derivable_pt id y).
-Intros; Apply derivable_pt_id.
-Assert H3 : (y:R)``a<=y<=x``->(continuity_pt id y).
-Intros; Apply derivable_continuous; Apply derivable_id.
-Assert H4 : (y:R)``a<y<x``->(derivable_pt f y).
-Intros; Apply pr; Elim H4; Intros; Split.
-Assumption.
-Elim H1; Intros; Apply Rlt_le_trans with x; Assumption.
-Assert H5 : (y:R)``a<=y<=x``->(continuity_pt f y).
-Intros; Apply H; Elim H5; Intros; Split.
-Assumption.
-Elim H1; Intros; Apply Rle_trans with x; Assumption.
-Elim H1; Clear H1; Intros; Elim H1; Clear H1; Intro.
-Assert H7 := (TAF_gen f id a x H4 H2 H1 H5 H3).
-Elim H7; Intros; Elim H8; Intros; Assert H10 : ``a<x0<b``.
-Elim x1; Intros; Split.
-Assumption.
-Apply Rlt_le_trans with x; Assumption.
-Assert H11 : ``(derive_pt f x0 (H4 x0 x1))==0``.
-Replace (derive_pt f x0 (H4 x0 x1)) with (derive_pt f x0 (pr x0 H10)); [Apply H0 | Apply pr_nu].
-Assert H12 : ``(derive_pt id x0 (H2 x0 x1))==1``.
-Apply derive_pt_eq_0; Apply derivable_pt_lim_id.
-Rewrite H11 in H9; Rewrite H12 in H9; Rewrite Rmult_Or in H9; Rewrite Rmult_1r in H9; Apply Rminus_eq; Symmetry; Assumption.
-Rewrite H1; Reflexivity.
-Assert H2 : x==a.
-Rewrite <- b0 in H1; Elim H1; Intros; Apply Rle_antisym; Assumption.
-Rewrite H2; Reflexivity.
-Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H2 H3) r)).
-Qed.
-
-(* La primitive est unique a une constante pres *)
-Lemma antiderivative_Ucte : (f,g1,g2:R->R;a,b:R) (antiderivative f g1 a b) -> (antiderivative f g2 a b) -> (EXT c:R | (x:R)``a<=x<=b``->``(g1 x)==(g2 x)+c``).
-Unfold antiderivative; Intros; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _; Exists ``(g1 a)-(g2 a)``; Intros; Assert H3 : (x:R)``a<=x<=b``->(derivable_pt g1 x).
-Intros; Unfold derivable_pt; Apply Specif.existT with (f x0); Elim (H x0 H3); Intros; EApply derive_pt_eq_1; Symmetry; Apply H4.
-Assert H4 : (x:R)``a<=x<=b``->(derivable_pt g2 x).
-Intros; Unfold derivable_pt; Apply Specif.existT with (f x0); Elim (H0 x0 H4); Intros; EApply derive_pt_eq_1; Symmetry; Apply H5.
-Assert H5 : (x:R)``a<x<b``->(derivable_pt (minus_fct g1 g2) x).
-Intros; Elim H5; Intros; Apply derivable_pt_minus; [Apply H3; Split; Left; Assumption | Apply H4; Split; Left; Assumption].
-Assert H6 : (x:R)``a<=x<=b``->(continuity_pt (minus_fct g1 g2) x).
-Intros; Apply derivable_continuous_pt; Apply derivable_pt_minus; [Apply H3 | Apply H4]; Assumption.
-Assert H7 : (x:R;P:``a<x<b``)(derive_pt (minus_fct g1 g2) x (H5 x P))==``0``.
-Intros; Elim P; Intros; Apply derive_pt_eq_0; Replace R0 with ``(f x0)-(f x0)``; [Idtac | Ring].
-Assert H9 : ``a<=x0<=b``.
-Split; Left; Assumption.
-Apply derivable_pt_lim_minus; [Elim (H ? H9) | Elim (H0 ? H9)]; Intros; EApply derive_pt_eq_1; Symmetry; Apply H10.
-Assert H8 := (null_derivative_loc (minus_fct g1 g2) a b H5 H6 H7); Unfold constant_D_eq in H8; Assert H9 := (H8 ? H2); Unfold minus_fct in H9; Rewrite <- H9; Ring.
-Qed.
-
 (* $\int_a^b f = -\int_b^a f$ *)
 Lemma NewtonInt_P4 : (f:R->R;a,b:R;pr:(Newton_integrable f a b)) ``(NewtonInt f a b pr)==-(NewtonInt f b a (NewtonInt_P3 f a b pr))``.
 Intros; Unfold Newton_integrable in pr; Elim pr; Intros; Elim p; Intro.